Set up an exponential function to model each real world situation then use it to find the requested values.
The half-life of Carbon 14 is about 5700 years. You begin with 800 grams of this isotope.
1) Function:
2) How much Carbon 14 is there after: a) 11,400 years b) 45,000 years
f(t) = 800(1/2)^(t/5700)
You can see that every time t increases by 5700, you have another power of 1/2, or a half-life.
So, using that, just plug in your values for t.
You can get an estimate by noting that
11400 = 2*5700
45000 ≈ 8*5700
To set up an exponential function to model the decay of Carbon 14, we can use the general form of an exponential function:
N(t) = N₀ * e^(kt)
Where:
- N(t) is the amount of Carbon 14 remaining after time t
- N₀ is the initial amount of Carbon 14
- k is the decay constant
- e is the base of the natural logarithm (~2.71828)
1) Function:
Since the half-life of Carbon 14 is about 5700 years, we can use this information to find the decay constant, k.
Half-life equation:
N(t) = N₀ * (1/2)^(t/h)
Where:
- h is the half-life
By substituting t = h and N(t) = N₀/2 into the equation, we can solve for k:
N₀/2 = N₀ * (1/2)^(h/k)
Simplifying the equation, we get:
(1/2)^(h/k) = 1/2
(1/2)^(h/k) = (1/2)^1
Since the bases are equal, the exponents must also be equal:
h/k = 1
Therefore, we have:
k = h
Now we can substitute this value of k in the exponential function:
N(t) = N₀ * e^(kt)
N(t) = N₀ * e^(ht)
For the given situation, let's set N₀ = 800 grams.
2) a) To find the amount of Carbon 14 after 11,400 years, we can use the exponential function:
N(t) = N₀ * e^(ht)
Substituting N₀ = 800 grams and t = 11,400 years:
N(11,400) = 800 * e^(5700 * 11,400)
You can use a calculator to calculate the exponential expression.
2) b) To find the amount of Carbon 14 after 45,000 years, we can again use the exponential function:
N(t) = N₀ * e^(ht)
Substituting N₀ = 800 grams and t = 45,000 years:
N(45,000) = 800 * e^(5700 * 45,000)
You can use a calculator to calculate the exponential expression.